Split real form

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra and let ${\mathfrak{g}}_0$ be a real form of $\mathfrak{g}$ (Real form of a complex Lie algebra). Then ${\mathfrak{g}}_0$ is a split real form (or normal real form) of $\mathfrak{g}$ if it contains a Cartan subalgebra ${\mathfrak{h}}_0$ (Cartan subalgebra) such that every adjoint operator ${\mathrm{ad}}_H$, $H\in {\mathfrak{h}}_0$, is diagonalizable over $\mathbb{R}$: that is, for each $H\in {\mathfrak{h}}_0$ the characteristic polynomial of ${\mathrm{ad}}_H$ splits into linear factors over $\mathbb{R}$, equivalently ${\mathfrak{g}}_0$ has a basis in which all ${\mathrm{ad}}_H$ are simultaneously diagonal with real eigenvalues. Since a split real form contains a Cartan subalgebra whose adjoint action is diagonalizable over $\mathbb{R}$, the complex rank of $\mathfrak{g}$ equals the real rank ${\dim }_{\mathbb{R}}{\mathfrak{h}}_0$ of this form. Existence and uniqueness up to real isomorphism are proved in Existence and uniqueness up to isomorphism of the split real form.